MATH SOLVE

2 months ago

Q:
# Find the radius of convergence, R, of the series. ∞ (−1)n (x − 6)n 4n + 1 n = 0 R = Find the interval, I, of convergence of the series. (Enter your answer using interval notation.) I =

Accepted Solution

A:

I'm guessing the series is[tex]\displaystyle\sum_{n=0}^\infty\frac{(-1)^n(x-6)^n}{4n+1}[/tex]Use the ratio test: we have[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(-1)^{n+1}(x-6)^{n+1}}{4n+5}}{\frac{(-1)^n(x-6)^n}{4n+1}}\right|=|x-6|\lim_{n\to\infty}\left|\frac{4n+1}{4n+5}\right|=|x-6|[/tex]The series converges for[tex]|x-6|<1\implies-1<x-6<1\implies5<x<7[/tex]which has a radius of convergence of 1.